Lemma

It follows that in this case the category of collections Coll(V)Coll(V) is a cofibrantly generated model category where a morphisms is a fibration or weak equivalence if it is so degreewise in VV, respectively.

Lemma

A VV-operad is called Σ\Sigma-cofibrant if its underlying collection is cofibrant in the above model stucture

A VV-operad PP is called reduced if P(0)P(0) is the tensor unit, P(0)=IP(0) = I. A morphism of reduced operads is one that is the identity on the 0-component.

Theorem

Then there exists a cofibrantly generated model structure on the category of VV-operads, in which a morphism P→QP \to Q is a weak equivalence (resp. fibration) precisely if for all n≥0n \geq 0 the morphisms P(n)→Q(n)P(n) \to Q(n) are weak equivalences (resp. fibrations) in VV.

The homotopy algebras over a simplicial/topological operad as defined by Boardman and Vogt (see references below), are algebras for cofibrant replacements of these operads in this model structure. This is essentially the statement of theorem 4.1 in (Vogt)

Remark

The category of CC-coloured operads is itself the category of algebras over a non-symmetric operad. See coloured operad for more. Thus the above theorem provides conditions under which CC-coloured operads carry a model structure in which fibrations and weak equivalences are those morphisms of operads P→QP \to Q that are degreewise fibrations and weak equivalences in ℰ\mathcal{E}.

Terminology

We shall from now on call an operad PPcofibrant if the morphism IC→PI_C \to P from the initial CC-coloured operad has the left lifting property against degreewise acyclic fibrations of operads (irrespective of whether the above conditions for the existence of the model structure hold).

If P→QP \to Q is a Σ\Sigma-cofibration between well-pointed Σ\Sigma-cofibrant CC-coloured operads, then the induced map W(H,P)→W(H,Q)W(H,P) \to W(H,Q) is a cofibration of cofibrant CC-coloured operads.